I will discuss recent numerical simulations of fluid flows with moving boundaries and interfaces which have been made possible thanks to the use of computational geometry. The talk will be organized in two parts: * I will first present early lower order representations of deforming interfaces like the Volume of Fluid Method, and the cut cell method for moving solid boundaries. I will also quickly present recent developments with these methods. * I will then present higher order geometric representations of deforming surfaces and applications to spectrally accurate simulations of flows ranging from red blood cells to solid rocket boosters. Convergence and stability of these recent methods will be discussed.